[GamingPuzzled]

No κ claim. These are screening ceilings on Cao Yifan's #4610 round-three productive-helper word, which the current #5413 caller still uses. They show where the remaining headroom is not.

Lever	Result
Productive re-ranking (finish rank-4/7/8 helpers as rank 9)	always loses (strict subadditivity of x^(1−C))
Longer controllers and query thinning	already closed by #4658/#4659/#4664/#4666/#4674
One climb per register; free reorder	ceilings 0.00715 and 0.00711 (not reachable)
Reorder within the exact commutation poset, dimension-only	0.00331 / 0.00323, but does not lift to real subspaces (332 clashes)
Reorder within the poset + re-frame, exact subspace LP (3.03M variables)	≤ 1.17 cost units per local, κ ≤ ≈ 0.003107
Exact subspace MIP, restricted and medium domains	the current word is optimal (gap 0)
Non-commuting swaps with exact correction ADDs	0 gains

Why. Each register climbs in 3–9 steps because a scalar ADD needs both operands at the identical frame. The helper frames forced in every valid reordering already cost 3458.82 of 3464.74. Further gains need a redesigned helper factorization, not reordering. Together with #5462 (the owned-relation optimum), this maps the edits on this word that are exhausted.

Everything is in research/frontier-word-reorder-ceilings/: README, R12/R13 notes, a working digest of the accounting, tools, logs and solutions. Most bounds are floating-point screens; the README labels which are exact. The positive control realize.py rebuilds the current word exactly and passes both native interpreters on all 632 columns.

AI disclosure: prepared by Joel Pulikkan (GamingPuzzled) with Claude Opus 5.5 (Anthropic) via Claude Code. Apache-2.0.

🤖 Generated with Claude Code